Three weddings and a funeral

This is a continuation of the series on submetries. First, an example: the distance function to a closed convex set is a submetry from onto . Note that the best regularity we can expect from such distance function is , the Lipschitzness of first derivatives. The second derivative does not exist at the points where level curves make the transition from straight to circular.

Distance function is a submetry

Now back to an attempt to introduce duality between immetries (=isometric embeddings) and submetries. Recall that the Lipschitz dual of a pointed metric space is the set of all Lipschitz functions such that . Given the Lipschitz norm, becomes a Banach space. Any closed ball can be written as
(*) .
Indeed, is obvious, and the reverse inclusion follows by considering .

It’s natural to use (*) to relate immetries to submetries, because the former are defined by and the latter by .

In the previous post I considered four implications:

If is an immetry, then is a submetry

If is a submetry, then is an immetry

If is an immetry, then is a submetry

If is a submetry, then is an immetry

I proved (1) already. Implication (2) is also easy to prove: if is a submetry, then for every , because pairs of points that (almost) realize can be lifted through .

Here is a proof of (4). If is a submetry, then . We can use this in (*) to obtain, for any and ,.
The latter set is nothing but , which means is an immetry.

I already noted that (3) fails for general metric spaces, the inclusion being a counterexample. However, this counterexample is not impressive, because is dense in . One could still hope that (3) holds for proper metric spaces. By definition, a metric space is proper if every closed ball is compact. (Or, equivalently, every bounded sequence has a convergent subsequence.) In metric geometry properness is a very common assumption which excludes incomplete spaces and infinite-dimensional spaces. Its relevance to submetries can be seen from their definition . It requires the image to be closed, which is not very likely to happen unless is compact. (No such issues arise on the immetry side, because is always closed.)

However, it turns out (3) is false even for compact spaces. The counterexample was already present on this blog:

These aren't the quotients I'm looking for

Indeed, the Lipschitz norm of a function defined on an interval is equal to the essential supremum of . The composition with the metric quotient map given above preserves . Hence, the induced map is an immetry.